Antenna and Radio Waves (vertical slice)

What is modelled

A vertical thin-wire dipole of length L is fed at its centre at a height of 4.4 m. A whip antenna of adjustable length sits on a radio 25 m away and is tilted by hand. The transmitter radiates the power you set; the receiver collects part of it. The ground slice, the green pattern surface, the LEDs and all graphs come from the same formulas.

Equations

• Wavelength: λ = c / f, wave number k = 2π/λ. The animation is slowed by a factor of about 10⁸; the spacing of the wave fronts is the true wavelength.
• Current on a thin centre-fed wire (standing wave): I(z) = I0 sin[k(L/2 - |z|)].
• Far field: Eθ = j η I0 e-jkr F(θ) / (2π r) with F(θ) = [cos(kL/2 cosθ) - cos(kL/2)] / sinθ and η = 376.73 Ω. For L = λ/2 this is the familiar cos(π/2 cosθ)/sinθ.
• Radiated power and radiation resistance (referred to the current antinode): Rr = (η/2π) ∫0π F2 sinθ dθ; directivity D = 2 Fmax2 / ∫ F2 sinθ dθ (1.64 = 2.15 dBi for a half-wave dipole). Input resistance at the feed: Rin = Rr / sin2(kL/2).
• Reactance (transmission-line approximation of a thin dipole): Xin = -Zc cot(kL/2) with Zc = 120 (ln(L/a) - 1), wire radius a = 3 mm. It is zero at L = λ/2 (resonance), negative (capacitive) for a shorter whip.
• Link (Friis): Pr = Pt Gt(θt) Gr(θr) · PLF · q · (λ / 4πd)2, where G = D F2(θ)/Fmax2, PLF is the polarisation factor (squared cosine between the field and the whip projected on the wave front) and q = 4 Rin RL / [(Rin + RL)2 + X2] is the impedance-mismatch factor for a 50 Ω receiver.
• Effective length of the whip (receive): leff = ∫ I dz / Iin = (2/k) tan(kL/4), and effective aperture Ae = η leff2 / (4 Rin) = λ2 D / 4π (reciprocity).

Checks run in the page

• Half-wave dipole: Rr = 73.1 Ω and D = 1.64 from numerical integration against textbook values; short dipole limit Rin = 20π2(L/λ)2.
• Power conservation: the gain integrated over the whole sphere equals 4π for the transmitter length you chose.
• Reciprocity: the aperture from the circuit (integrating the current numerically) equals the aperture from the directivity, for the whip length you chose.
• Received power by two independent routes: the Friis formula and field strength to open-circuit voltage to a 50 Ω load.

Honest simplifications

• Thin wire with an assumed sinusoidal current. A real antenna has a slightly different current (finite thickness, end effects), which shifts the resonant length a few percent shorter than λ/2 and changes the reactance; the cot(kL/2) formula is only reliable near resonance.
• Free space: no ground reflection, no multipath, no obstacles. Real VHF links have ground reflections that cause deep interference notches with distance and height.
• The transmitter is assumed lossless and perfectly matched, so all of P_t is radiated. The receiver is a 50 Ω resistor with no matching network; the -90 dBm noise floor and the six signal bars are illustrative values, not those of a specific radio.
• The wave-front slice shows the far-field expression everywhere; it is not valid within about one wavelength of the antenna (near fields are not modelled).

• Press Whip too short and then Whip too long (1.5 lambda) and watch the second graph: the received power peaks near a half wavelength and has dips and bumps beyond it, because the whip pattern splits into lobes.
• Press Whip lying sideways: the transmitted field is vertical, so a horizontal whip cannot follow it and the polarisation factor falls to zero.
• Press Full-wave transmitter or 1.5 lambda transmitter and see the polar pattern and the colour map grow extra lobes while the field along the line of sight changes.
• Drag the distance and read the third graph: the power falls as 1/r^2, 6 dB for every doubling of distance, until it meets the red noise floor.